I am studying for an upcoming Linear Algebra exam. I am going through the questions from an old exam the instructor gave out, and I have come to this problem:
Give an example of an operator on a complex vector space with characteristic polynomial $(z-2)^3 (z-3)^3$ and with minimal polynomial $(z-2)^3(z-3)^2$.
Now I know that the matrix for this operator must have three $2$'s and three $3$'s down the diagonal, and I know the minimal polynomial divides the characteristic, but I don't know much else. This is in the same chapter as Jordan form, so I think a solution might have to do with Jordan blocks, but I don't have enough intuition about those to get it.
Any help here? 🙂
Best Answer
$$ \left( \begin{array}{cccccc} 2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 0 & 0 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 0 & 0 & 3 \end{array} \right). $$